Study of $\rm{K}^{*0}$ and $\phi$ meson Production in proton+proton Collisions with ALICE at the LHC and Application of Non-extensive Statistics in High Energy Physics

In today's universe, all ordinary matter is made up of atoms and these atoms are made up of protons and neutrons (except hydrogen) surrounded by cloud of electrons. The protons and neutrons are made up of quarks and bound together by gluons. The strong interactions between quarks are described by the theory of Quantum Chromo Dynamics (QCD). The QCD has two important properties because of the non-abelian nature of the gauge bosons. At high energy or momentum transfer (Q$^2$) the strong coupling constant becomes so small as if the interacting partons are free which is known as asymptotic freedom. On the other hand, at lower-momentum transfer, the coupling constant becomes high and hence it becomes impossible to separate the individual quarks to observe them free. This is because of the confining nature of strong interaction. These properties can lead to interesting phases of matter depending upon the temperature and/or density as the quarks and gluons become the relevant degrees of freedom instead of hadrons under such extreme circumstances. It has been speculated that a few micro-seconds after the Big-bang when the temperature was ~100 MeV (10$^{12}$ K) the universe was filled with quarks and gluons which are not free but deconfined. This phase is known as quark-gluon plasma (QGP). To create and study the QGP phase is the main motivation behind the ultra-relativistic heavy-ion collision experiments, where extreme temperatures/densities can be achieved. The experimental search for the de-confined state of quarks and gluons got a boost from the year 2000 at the Relativistic Heavy Ion Collider (RHIC) at BNL, where Au ions collide at $\sqrt{s_{NN}}$ = 20 -- 200 GeV and thereafter in the Large Hadron Collider (LHC) at CERN. All four RHIC experiments (BRAHMS, PHOBOS, PHENIX, and STAR) have discovered a new phase of nuclear matter that exhibits the properties of the strongly interacting, nearly perfect liquid. At RHIC, various signatures such as jet quenching, azimuthal anisotropy, J/$\psi$ suppression, and strangeness enhancement indicated a creation of a deconfined state of quarks and gluons known as QGP. A Large Ion Collider Experiment (ALICE) at the LHC is specifically designed for the creation of QGP and study its properties. The system formed in the heavy ion collisions undergoes a phase transition from a state of quark-gluon plasma to a phase of hadron gas. During this phase transition, the boundary at which the inelastic processes cease known as the chemical freeze-out and similarly at kinetic freeze-out boundary the elastic processes cease. The phase between chemical freeze-out and kinetic freeze-out boundaries is known as the hadronic phase. The lifetime of the fireball created in high energy collisions is comparable to the lifetime of resonance particles such as $K^{*}(892)^0$ and $\phi(1020)$, which have lifetimes ~4.16 $\pm$ 0.05 fm/${\it c}$ and 46.3 $\pm$ 0.4 fm/${\it c}$. By studying the resonance particle we can probe the hadronic phase produced in high energy hadronic and nuclear collisions. The $\phi(1020)$ meson is one of the lightest vector mesons consisting of ($\mathrm{s\bar{s}}$), helps in the study of the strangeness production. Similarly, $K^{*}(892)^0$ is also a vector meson having a similar mass to $\phi(1020)$ but differs in strangeness content by one unit. A combined study of both the mesons helps to probe the strangeness production mechanism. Here, we study the minimum bias measurements of $K^{*}(892)^0$ and $\phi(1020)$. These results include: the transverse momentum spectra ($\rm{p}_T$), mean transverse momentum, differential yields and also particle ratios with respect to stable particles. These results also provide an input to tune the perturbative-Quantum Chromodynamics (pQCD) inspired Monte Carlo (MC) event generators such as PYTHIA, PHOJET, and EPOS. Furthermore, the measurements in minimum bias pp collisions at $\sqrt{\mathrm{s}}$ = 8~TeV~reported in this thesis serve as reference data to study nuclear effects in proton-lead (p--Pb) and Pb-Pb collisions. High multiplicity pp-collisions at the LHC give us a possible opportunity to study matter under extreme conditions {\it i.e.} at high temperature and/or energy density and hence to explore the possibility of a formation of a QGP droplet in pp-collisions. The kinetic freeze-out surface of identified particles are determined by studying the transverse momentum spectra ($p_{\rm T}$) of the produced particles in high energy collisions. Here, we analyse the transverse momentum ($p_{\rm T}$)-spectra as a function of charged-particle multiplicity at midrapidity ($|y| < 0.5$) for various identified particles in proton-proton collisions at $\sqrt{s}$ = 7 TeV using Boltzmann-Gibbs Blast Wave (BGBW) model and thermodynamically consistent Tsallis distribution function. We obtain the multiplicity dependent kinetic freeze-out temperature ($T_{\rm kin}$) and radial flow ($\beta$) of various particles after fitting the $p_{\rm T}$-distribution with BGBW model. Here, $T_{\rm kin}$ exhibits mild dependence on multiplicity class while $\beta$ shows almost independent behaviour. The information regarding Tsallis temperature and the non-extensivity parameter ($q$) are drawn by fitting the $p_{\rm T}$-spectra with Tsallis distribution function. The extracted parameters of these particles are studied as a function of charged particle multiplicity density ($dN_{ch}/d\eta$). In addition to this, we also study these parameters as a function of particle mass to observe any possible mass ordering. All the identified hadrons show a mass ordering in temperature, non-extensive parameter and also a strong dependence on multiplicity classes, except the lighter particles. It is observed that as the particle multiplicity increases, the $q$-parameter approaches to Boltzmann-Gibbs value ($q \sim 1$) indicating the tendency of the system to approach a thermal equilibrium. The observations are consistent with a differential freeze-out scenario of the produced particles. To explain the identified particle spectra in heavy-ion collisions, incorporation of radial flow in Tsallis statistics is necessary and thus assuming $(q-1)$ close to zero, a Taylor series expansion of Tsallis distribution function in powers of ($q-1$) is provided. This helps in studying the degree of deviation of transverse momentum spectra and other thermodynamic quantities from a thermalized Boltzmann distribution. After checking thermodynamic consistency, we provide analytical results for the Tsallis distribution in the presence of collective flow up to first order in ($q-1$). The theoretical expectations are compared with the experimental data. Similarly, the speed of sound ($c_s$) is studied to understand the hydrodynamical evolution of the matter created in heavy-ion collisions. The quark-gluon plasma (QGP) formed in heavy-ion collisions evolves from an initial QGP to the hadronic phase via a possible mixed phase. Due to the system expansion in a first order phase transition scenario, the speed of sound reduces to zero as the specific heat diverges. We study the speed of sound for systems, which deviate from a thermalized Boltzmann distribution using non-extensive Tsallis statistics. In the present work, we calculate the speed of sound as a function of temperature for different $q$-values for a hadron resonance gas. We observe a similar mass cut-off behaviour in the non-extensive case for $c^{2}_s$ by including heavier particles, as is observed in the case of a hadron resonance gas following equilibrium statistics. Also, we explicitly show that the temperature where the mass cut-off starts to vary with the $q$-parameter which hints at a relation between the degree of non-equilibrium and the limiting temperature of the system. It is shown that for values of $q$ above approximately 1.13, all criticality disappears in the speed of sound, i.e. the decrease in the value of the speed of sound, observed at lower values of $q$, disappears completely. Again, thermodynamic observables such as specific heat and isothermal compressibility are very useful in quantifying the nature of the phase transition and to obtain the equation of state (EOS) of the produced matter. Here, we have studied the isothermal compressibility ($\kappa_T$) as a function of temperature, baryon chemical potential and centre-of-mass energy ($\sqrt{s_{NN}}$) using hadron resonance gas (HRG) and excluded-volume hadron resonance gas (EV-HRG) models. A mass cut-off dependence of isothermal compressibility has been studied for a physical resonance gas. Further, we study the effect of heavier resonances ($>$ 2 GeV) on the isothermal compressibility by considering the Hagedorn mass spectrum, ${\rho}(m)\sim{\exp(bm)}/{(m^2+m_0^2)^{5/4}}$. Here, the parameters, $b$ and $m_0$ are extracted after comparing the results of recent lattice QCD simulations at finite baryonic chemical potential. We find a significant difference between the results obtained in EV-HRG and HRG models at higher temperatures. The inclusion of the Hagedorn mass spectrum in the partition function for hadron gas has a large effect at a higher temperature. A higher mass cut-off in the Hagedorn mass spectrum takes the isothermal compressibility to a minimum value, which occurs near the Hagedorn temperature ($T_H$). Furthermore, it is always exciting to explore possible applications of non-extensivity in condensed matter physics- a cross-area application. Here, we use Tsallis thermodynamically consistent distribution in studying the thermal effective mass and specific heat of various metals. We have found an excellent agreement with experimental findings. These works form the basis of the present thesis.